There is an important relationship between the PH and PrcH cycle bases for \(F_{} C\)  and \(D,\)  given by the following theorem. The part of Theorem REF  that applies to PH was proved in [1]}; the dual component for PrcH was introduced in [2]}. A matrix with exactly one non-zero entry in each row and each column is a (generalized) matching matrixA matching matrix is a generalized matching matrix with entries in \(\lbrace 0,1\rbrace \) . We deal only with generalized matching matrices in this text and will, by abuse of terminology, drop the qualifier “generalized.”..
[Global duality for generators, [1]}, [2]}]

Suppose that \(E\)  is a 0-graded invertible upper triangular matrix and \(M= E^{-1} DE\)  is a matching matrix. Then,
